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INTEREST RATES 1: WHAT ARE INTEREST RATES?
AP Faure1
ABSTRACT
This is the first in a series of seven papers on interest rates and it covers the
basic terms and information required for a fuller understanding of the significance
of interest rates: the instruments that interest rates apply to, the bank interest
margin which plays an important transmission role in monetary policy, time value
of money, types, relationship between rates and prices, and other relevant
issues. The seven papers cover: (1) what are interest rates?; (2) relationship of
interest rates; (3) composition of interest rates; (4) interest rate discovery; (5)
bank liquidity & interest rate discovery; (6) role of interest rates; (7) an optimal
rate of interest: the natural rate.
JEL Classification: E 40, E43, E44, E50, E51, E52, E58
Keywords: interest rates, money market, monetary policy, money, financial
markets
1. INTRODUCTION
Before a study of this material is undertaken, it is important to have an
understanding of the financial system  which provides the context of interest
rates. We suggest “Financial System: An Introduction” which is available free at
http://bookboon.com/en/financialsystemanintroductionebook. For those who
are familiar with the system, we offer a brief reminder below.
Interest rates are the reward paid by a borrower (debtor) to a lender (creditor) for
the use of money for a period, and they are expressed in percentage terms per
annum (pa), for example, 6.525% pa, in order to make them comparable. Interest
rates are also quite often referred to as the price of money. This is not helpful.
One should rather refer to interest rates as being the rates (there are various)
payable on debt and deposit obligations (aka instruments and securities) by the
borrowers to the lenders, and that the prices of the debt and deposit obligations
are derived from the cash flows payable on the obligations in the future  by
discounting the cash flows by the rates payable.
1 Rhodes University
2
Upfront we offer a significant statement: shortterm interest rates are not
determined by supply and demand; they are controlled by the central bank (and
there is an especially good reason for this), and all other interest rates are a
function of current shortterm rates and expectations as to where they will be in
the future. Supply and demand forces do enter the equation  to the extent that
the central back reacts to these forces with its administrativelydetermined
interest rate, the policy interest rate (PIR). They do play a role in the rate
determination on longer term obligations, but the PIR remains the anchor. We will
return to these issues many times.
The term interest rate/s can be quite confusing to those unfamiliar with the
financial markets. There are many different interest rates; a few examples: call
deposit rates, term deposit rates, repurchase agreement (repo) rates, base rates,
policy rates, bank rates, government bond rates, corporate bond rates,
negotiable certificates of deposit (NCD) rates, Treasury bill (TB) rates, corporate /
commercial paper (CP) rates, fixed interest rates, floating interest rates, discount
rates, coupon rates, real rates, nominal rates, effective rates, riskfree rates, and
so on.
Confusing? Yes, but they are all related and there is a way demystify the
terminology. This is the aim of this text. It also elucidates the significant role of
interest rates in the economy. We begin with: interest rates apply only to debt
and deposit instruments (there are a few exceptions, such as preference shares).
To comprehend this, we need to provide a synopsis of the financial system. This
is provided next. The organisation of this text is as follows:
Financial system: a synopsis.
Debt and deposits.
The bank margin.
Rate of interest.
Time value of money.
TVM and compound interest.
Effective rate.
Coupon rate.
Price of a security: the principle.
Price of a security: multiple future cash flows and yield to maturity.
Other issues and terminology related to interest rates.
2. FINANCIAL SYSTEM: A SYNOPSIS
We present Figure 1 as the backdrop to this brief discussion. Perusal of the
figure will reveal:
First: Ultimate borrowers issue financial securities, meaning that they borrow
funds and issue evidences thereof (aka securities, IOUs, instruments,
obligations, etc). There are only two: debt and shares / equities. The ultimate
3
lenders lend their excess funds, meaning that they purchase securities
(evidences of debt and shares). The ultimate lenders and borrowers are
comprised of the same four sectors of the economy, as indicated. Some of them
are lenders and borrowers at the same time (for example, government), but
generally they are one or the other.
Second: Financial intermediaries interpose themselves between the ultimate
lenders and borrowers by offering useful financial services. They have assets
(buy securities) and liabilities (issue their own securities to fund their assets). The
main financial intermediaries are:
Banks (central bank and private sector banks): They buy debt securities
and issue securities known as certificates of deposit (CDs) which are
negotiable (ie marketable, called NCDs) or nonnegotiable (NNCDs). They
are overwhelmingly of a shortterm nature. Note: The central bank’s
liabilities are not termed as such; we call them CDs for the sake of
simplicity.
Investment vehicles: They buy debt and shares and issue what may be
called “participation interests” (PIs). Other names are membership
interests and units.
Third: Debt securities are divided into longterm (LT) securities and shortterm
(ST) securities, and they are either marketable debt (MD) or nonmarketable debt
(NMD), ie the financial system has LTMD, LTNMD, STMD and STNMD.
Marketable debt is marketable because secondary markets exist for them.
Fourth: Shares are issued by companies and are marketable (MS) or non
marketable (NMS). Debt and shares are issued in primary markets and traded in
secondary markets, such as a stock exchange, making them marketable.
INVESTMENT VEHICLES
•Contractual Intermediaries
•Retirement fund s
•Insurers
•Collective Inv Schemes
•Unit trust s
•Exc hange traded f unds
•Alternative Investments
•Hedg e funds
•Private equity funds
Deb t & share s ecuriti es
Deposit
Securities (CDs)
Deposit securities (certificates of deposit –CDs)
Participatio n
interest (PI)
securities
Surplus funds
CENTRAL
BANK
BANKS
BANKS
Interbank
debt
Interbank
debt
ULTIM ATE
BORROWE RS
(deficit economic
units)
HOUSEHOLD
SECTOR
CORPORA TE
SECTOR
GOVERNMENT
SECTOR
FOREIGN
SECTOR
ULTIMATE
LENDE RS
(surplus economic
units)
HOUSEHO LD
SECTOR
CORPORATE
SECTOR
GOVERNMENT
SECTOR
FOREIGN
SECTOR
Deb t & share s ecuriti es
Deb t & share s ecuriti es
Figure 1.1: Financial system
An example will render the above comprehensible: A bank makes a mortgage
loan to you to buy a house, and funds it by issuing CDs to a company:
4
You are an ultimate borrower (a member of the household sector) and you
issue an LTNMD (an IOU), meaning you owe the bank.
The bank buys your LTNMD and issues CDs.
The company (ultimate lender, a member of the corporate sector) buys the
CDs.
Another way of seeing the financial system: There are six elements:
First: Ultimate lenders (= surplus economic units) and ultimate borrowers (=
deficit economic units), ie the nonfinancial economic units that undertake the
lending and borrowing process. The ultimate lenders lend to borrowers either
directly or indirectly via financial intermediaries, by buying the securities they
issue.
Second: Financial intermediaries which intermediate the lending and borrowing
process. They interpose themselves between the lenders and borrowers, and
earn a margin for the benefits of intermediation (including lower risk for the
lenders). They buy the securities of the borrowers and issue their own to fund
these (and thereby become intermediaries).
Third: Financial instruments (or securities, obligations, assets), which are created
/ issued by the ultimate borrowers and financial intermediaries to satisfy the
financial requirements of the various participants. These instruments may be
marketable (e.g. Treasury bills) or nonmarketable (e.g. retirement annuities).
There are two categories and two subcategories:
Ultimate financial securities (issued by ultimate borrowers):
o Debt securities.
o Share (aka stock / equity) securities.
Indirect financial securities (issued by financial intermediaries):
o Deposit securities, aka certificates of deposit (CDs) (issued by
banks).
o Participation interests (PIs) (issued by investment vehicles).
Fourth: Creation of money (= bank deposits; bank notes are also deposits) by
banks when they satisfy the demand for new bank credit. This is a unique feature
of banks. Central banks have the tools to control money growth, which they do
primarily to tame inflation.
Fifth: Financial markets, ie the institutional arrangements and conventions that
exist for the issue and trading (dealing) of the financial instruments. The financial
markets are:
Money market (all STMD, STNMD and CDs), in other words the entire
shortterm debt and deposit market, marketable and nonmarketable.
Bond market (all LTMD), in other words the marketable part of the long
term debt market.
Share / stock / equity market (all MS).
5
Foreign exchange market (the market for the exchange of currencies).
Participation interests markets (there are a number, e.g. units of unit
trusts, membership interest in a retirement fund).
Derivatives markets (forwards, futures, swaps, options, etc).
Sixth: Interest rate / price discovery, ie the establishment in the financial markets
of the rates of interest on debt and deposit instruments, and the prices of share
instruments.
As our interest in this text is interest rates and their discovery, we can ignore
shares and PIs, which do not carry interest (there are exceptions, such as
preferences shares, but we will ignore them in the interests of pedagogy). Thus,
we are left with debt and deposits, and their markets.
3. DEBT AND DEPOSITS
INVESTMENT
VEHICLES
CIs
CISs
AIs
CENTRAL
BANK
BANKS
BANKS
•Debt = NMD
• Debt = MD (TBs, bonds)
• Debt = MD (CP, bonds)
& NMD
•Debt
•Debt
Interbank
debt
Interbank
debt
• Debt = MD (bonds)
•CDs =
NCDs &
NNCDs
•CDs =
NCDs &
NNCDs
•Debt
•CDs
•CDs
MD = mark etable d ebt; NMD = nonmark etable debt ; CP = comm ercial paper; CDs = certificates of deposit (= deposits ); NCDs = nego tiab le c erti fic ates of de pos it; NNCD s = n onne go tiab le
certificates of deposit.
ULTIMATE
BORROWE RS
(deficit econo mic
units)
HOUSEHOLD
SECTOR
CORPORA TE
SECTOR
GOVERNME NT
SECTOR
FOREIGN
SECTOR
ULTIM ATE
LENDERS
(surplus econom ic
units)
HOUSEHOLD
SECTOR
CORPORATE
SECTOR
GOVE RNMENT
SECTOR
FOREIGN
SECTOR
Figure 1.2: Debt and deposit securities
Debt securities are evidences of debt issued by the borrower to the lender. The
lender may be a financial intermediary (bank, central bank or investment vehicle)
or an ultimate lender (one of the four sectors). As may be seen in Figure 2, the
following debt securities exist:
Household sector:
o MD (none, because they are not able to issue MD).
o NMD (examples: leases, mortgage loans, overdraft facilities
utilised).
Corporate sector:
o MD [commercial paper (CP), bankers’ acceptances (BAs),
promissory notes (PNs), corporate bonds].
o NMD (examples: leases, mortgage loans, overdraft facilities
utilised).
Government sector:
6
o MD [central government: Treasury bills (TBs), bonds].
o NMD (issued by the lower levels of government, for example, local
government bonds).
Foreign sector:
o MD [foreign commercial paper (CP), corporate bonds].
o NMD (none as only the large foreign corporate entities are able to
issue, and they issue MD).
Figure 2 shows: Deposit securities (CDs) are issued by banks to lenders, and the
funds are used to purchase debt securities, in the form of MD and NMD. The vast
majority are NMD, and specifically mortgage loans and overdraft facilities utilised.
As we have seen, there are two categories of CDs: NCDs and NNCDs. The vast
majority are NNCDs.
It is necessary at this time to make reference to the middle part of Figures 1 and
2, and enhanced in Figure 3: The interbank debt market (IBM). There are three
parts to the IBM:
Banktobank interbank market (b2b IBM). This is where interbank claims
and loans are settled, and this is effected over the accounts that banks are
required to have with the central bank.
Banktocentral bank interbank market (b2cb IBM). This represents the
reserve requirement amount, ie the requirement that banks are to hold a
certain proportion of their deposits with the central bank (in most
countries).
Central banktobank interbank market (cb2b IBM). This represents the
(usually overnight) loans made by the central bank to the banks for
monetary policy purposes.
CENTRAL BANK
BANKS
BANKS
Required reserves
(RR) (b2cb IBM)
IBM loans
(b2b IBM)
Loans to =
borrowed
reserv es
(BR)
(cb2b IBM)
Securities
ULTIM ATE
LENDERS
HOUSEHOLD
SECTOR
CORPORA TE
SECTOR
GOVERNMENT
SECTOR
FOREIGN
SECTOR
ULTIM ATE
BORROWERS
HOUSEHO LD
SECTOR
CORPORA TE
SECTOR
GOVERNMENT
SECTOR
FOREIGN
SECTOR
OTHER
FINANCIAL
INTER MED IAR IES
Securities
Securities
Securities
Figure 1.3: Interbank market
This significant market and its role in interest rate determination will be elucidated
in detail later.
7
It will be evident that each ultimate borrower security [ie the different types of
NMD and MD (CP, BAs, PNs, TBs, corporate and government bonds)] carries a
different interest rate. Similarly, each NNCD and NCD carries a different rate of
interest. This also applies to the IBM. However, the story is different in the IBM in
that the genesis of interest rates is found here, and it is determined
administratively to a significant degree, as we will show later.
Ignoring the influence of the central bank on interest rates (and therefore on
inflation), each interest rate is dependent upon / influenced by:
Term to maturity. The rate increases as the term to maturity increases.
Risk profile of the issuer (borrower: ultimate borrower or bank). The rate
rises as the risk rises.
Marketability (all issues of securities take place in their primary markets,
but only certain securities have secondary markets  where they can be
sold or bought). The rate decreases as marketability increases.
One final point, which we have not indicated in Figures 1 and 2 (but rectify in
Figure 4), is the existence of direct financing. Not all lending and borrowing takes
place via the financial intermediaries. It can also occur directly (an example is a
company or wealthy individual buying bonds with excess funds). However, the
vast majority is undertaken via the financial intermediaries.
Securities
FINANCIAL
INTERMEDIARIES
Securities
Indirect f inan cing
Securities
Direct finan cing
ULTI MATE
BORROWE RS
(deficit economic
units )
HOUSE HOLD
SECTOR
CORPORA TE
SECTOR
GOVERNMENT
SECTOR
FOREIGN
SECTOR
ULTI MATE
LENDE RS
(surp lus eco nomic
units)
HOUSEHOLD
SECTOR
CORPORA TE
SECTOR
GOVE RNMENT
SECTOR
FOREIGN
SECTOR
Surplus funds
Surplus funds Surplus funds
Figure 1.4: Direct and indirect financing
4. THE BANK MARGIN
It is important at this stage to introduce the bank margin. In simple terms (see
Figure 5; we will elucidate later) banks intermediate the lending and borrowing
8
process, in the process transmuting NMD into NNCDs and NCDs. In essence,
they are creating liquidity and reducing risk for the lenders (buyers of CDs =
depositors) by taking on the information costs and providing diversification of
assets.
For this service they charge a “fee” in the form of a lower rate of interest earned
by the lender (the buyer of CDs) than they earn on the MD and NMD securities
purchased (ie their loans / credit). This difference is the bank margin (IE – IP),
and it is “sticky” in that it is jealously guarded by the banks as it represents a
major part of their profits. It is kept at a reasonable number by competition in the
banking sector.
ULTIMATE
LENDERS
(surplus econo mic
units)
HOUSEHOLD
SECTOR
CORPORA TE
SECTOR
GOVERNMENT
SECTOR
FOREIGN
SECTOR
ULTIM ATE
BORROWE RS
(deficit econo mic
units)
HOUSEHOLD
SECTOR
CORPORATE
SECTOR
GOVERNMENT
SECTOR
FOREIGN
SECTOR
BANKS
Asse ts Liab ilities
DepositsLoans
Debt securities Deposit securities
BANK MARGIN
IE  IP
Intere st paid (IP) Interest earned (IE)
INVES TMENT
VEHICLES
Figure 1.5: Bank margin
The bank margin is significant for another important reason: The central bank, by
influencing the cost of the banks’ liabilities (“at the margin”), via the policy interest
rate (PIR), are able to directly influence the banks’ lending rates. The benchmark
bank lending rate is called prime rate (PR). PR is the high profile rate one sees in
press announcements, and all bank lending rates are related to it. For example, a
mortgage loan may be granted to a small borrower at PR+1%, and to a large
borrower at PR2%. Its prominence in monetary policy will become clear later.
5. RATE OF INTEREST
In order to concretise the understanding of interest, we need to go back to
basics. The rate of interest is the price or fee paid by a borrower of money to the
lender for the use of the money for a period, divided by the amount borrowed.
The borrower is thus advancing consumption and paying for this privilege. From
the perspective of the lender, the price or fee charged is his / her compensation
for delaying consumption for the period of the loan.
Thus, seen simply, there are two elements to the rate of interest, the price or fee
paid and the amount loaned / borrowed. An example:
9
Fee / price paid = LCC2 100
Amount loaned / borrowed = LCC 1 000.
The interest rate (ir) is as follows:
ir = fee paid / loan amount
= 100 / 1 000
= 0.10 (or 0.10 LCC per one unit of LCC loaned)
= 10%.
It should be evident that the rate of interest is a ratio, ie the ratio of (in this
example) 100 / 1 000. Also, this is the rate of interest for the relevant period. As
said above, in practice interest rates are expressed in pa terms, in order to make
them comparable.
The term of the loan, the rate and the pa convention are important. For example,
if the loan is for 91 days (t), and the 10% rate is for this period, the amount
payable is LCC 100, but the rate is not 10% pa; it is (assuming the daycount
convention is 365 days):
Effective rate pa = ir x (365 / t)
= 0.1 x (365 / 91)
= 0.40110
= 40.11% pa3.
If the 10% rate is a pa rate, then the amount payable on a LCC 1 000 loan after
91 days is:
Interest payable (IP) = loan amount x (ir x t / 365)
= LCC 1 000 x (0.1 x 0.24932)
= LCC 24.93.
These numbers can be used to derive the ir pa:
ir pa = (IP / loan) x (365 / t)
= [(LCC 24.93 / LCC 1 000) x (365 / 91)]
= 0.1
= 10% pa.
Compounding and multiple regular future interest payments will be introduced
later. The above discussion hints at the fact that money has a value over time.
We cover this next.
2 LCC is the currency code for a fictitious currency, the corona, of fictitious country, Local
Country.
3 Not compounded for the sake of simplicity.
10
6. TIME VALUE OF MONEY
The time value of money (TVM) concept, a significant concept in economics and
finance, means that money has a value over time. It is founded on the notion that
money represents a command over goods and services (ie consumption), and
that if you delay consumption by lending part of your money supply to someone,
you will expect compensation, otherwise you would not lend the money. What’s
the point? Even if you were inclined to lend the money to a friend compensation
free, this is a foolish idea, and there is a sound reason for this: the future is
uncertain. There are two factors to consider in relation to the future: you cannot
be certain that you will receive the money loaned and / or the compensation
amount when they are due (= credit risk), and inflation may erode the value of the
money lent (= inflation risk). As we know, the compensation amount is called
interest.
Another way of looking4 at this concept is that LCC 1 received today is worth
more than LCC 1 received in the future. This of course is because the LCC can
be invested and its value enhanced by the rate of return, the interest amount.
This is the basic tenet of the TVM concept, ie money has a future value (FV) and
a present value (PV):
FV is PV plus interest.
PV is FV discounted at the relevant interest rate.
Another basic principle of the concept is that interest is compounded, ie interest
that is earned is reinvested, and an essential assumption here is that interest
earned is reinvested at the rate earned on the principal amount. The PVFV
concept is the foundation of all financial market mathematics.
From the previous section, we know that the principal amount (amount invested)
is the PV and the FV is the sum of the PV and the interest amount (IA) earned,
as follows.
FV = PV + IA.
This may be expressed as:
FV = PV + [PV x (ir x t / 365)]
= PV x [1 + (ir x t / 365)].
From this we are able to derive the PV formula:
PV = FV / [1 + (ir x t / 365)].
4 Saunders and Cornett (2001: 26).
11
Example: PV to FV:
PV = LCC 1 000 000
ir = 14% pa
t = 90 days
FV = PV x [1 + (ir x t / 365)]
= LCC 1 000 000 x [1 + (0.14 x 90 / 365)]
= LCC 1 000 000 x 1.03452055
= LCC 1 034 520.55.
Example: FV to PV:
FV = LCC 1 350 000
ir = 12% pa
t = 120 days
PV = FV / [1 + (ir x t / 365)]
= LCC 1 350 000 / [1 + (0.12 x 120 / 365)]
= LCC 1 350 000 / (1.0394521)
= LCC 1 298 761.20.
7. TMV AND COMPOUND INTEREST
Compound interest takes into account interest earned on interest and on the
principal amount (ie the original amount of the investment / borrowing). It
assumes always that the interest earned is reinvested at the original rate of
interest from as soon as it is paid.
A simple example may be useful: a LCC 1 million investment for 2 years at 13%
pa payable in arrears (see Figure 6).
Time line
T+0 T+1
year
T+2
years
13%
LCC 1 000 00 0
investment
13%
LCC 130 000
cash flow
LCC 130 000 cash flow
LCC 16 900 cash flow
Figure 1.6: Compound interest (cash flows)
The LCC 1 million investment earns interest at 13% pa twice (LCC 130 000), ie
at the end of each year. The first interest payment is also invested at 13% (the
assumption as explained) for the last year [yielding LCC 16 900 (130 000 x
12
0.13)]. Thus the value of the investment at the end of the period of 2 years (FV)
is:
LCC 1 000 000 + (2 x LCC 130 000) + LCC 16 900 = LCC 1 276 900.
The compound interest formula is:
FV = PV x (1 + ir / not)y.not
where
ir = interest rate pa
y = number of years
not = number of times interest is paid per annum.
Example:
PV = LCC 1 000 000
ir = 13% pa
y = 2
not = 1
FV = LCC 1 000 000 x (1 + 0.13 / 1)2x1
= LCC 1 000 000 x (1.13)2
= LCC 1 000 000 x (1.2769)
= LCC 1 276 900.00.
Another example:
PV = LCC 1 000 000
ir = 15% pa
y = 1
not = 12 (ie monthly)
FV = LCC 1 000 000 x (1 + 0.15 / 12)1x12
= LCC 1 000 000 x (1.0125)12
= LCC 1 000 000 x (1.16075452)
= LCC 1 160 754.52.
Yet another example:
PV = LCC 1 000 000
ir = 15% pa
y = 3
not = 2 (ie sixmonthly)
13
FV = LCC 1 000 000 x (1 + 0.15 / 2)3x2
= LCC 1 000 000 x (1.075)6
= LCC 1 000 000 x (1.54330153)
= LCC 1 543 301.53.
The PV of an investment may be derived from the FV:
PV = FV / (1 + ir / not)y.not.
An example: What amount must be invested now (PV) at 12% pa compounded
semiannually to end up at LCC 1 million in 3 years’ time? The answer is:
PV = LCC 1 000 000 / (1 + 0.12 / 2)3.2
= LCC 1 000 000 / (1.06)6
= LCC 1 000 000 / 1.41851911
= LCC 704 960.54.
This is a significant formula in economics and finance. Borrowings and
investments have future cash flows (FVs). This formula enables one to calculate
the price (= PV) of an investment with future cash flows. Note that the interest
rate is part of the denominator, which means that when a rate rises, the price
(PV) of the investment falls. The converse obviously holds.
Note also that this formula is applied differently in the case of financial
instruments with multiple regular cash flows (FVs). As we will show later, each
cash flow is discounted to PV and then added.
8. EFFECTIVE RATE
Rates of interest pa in the financial markets are quoted with the interest
frequency stated. These rates are referred to as the nominal rates. For example,
a rate may be quoted as 13.5% pa with interest payable monthly, or a rate may
be quoted as 12% pa with interest payable quarterly.
The terminology used in the market for these two rates are 13.5% nacm (nominal
annual compounded monthly) and 12% nacq (nominal annual compounded
quarterly). In the case where interest is payable sixmonthly and at the end of a
year, the terminology would be nacs (nominal annual compounded semi
annually) and naca (nominal annual compounded annually).
In order to compare these rates, the term effective rate is applied. Nominal rates
are converted to effective rates with the use of the following formula:
ire = [(1 + irn / t)t – 1]
where
14
ire = effective rate
irn = nominal rate
t = number of interest periods per annum.
An example: A 12% nacm rate converts to an effective rate as follows:
ire = (1 + ir
n / t)t – 1
= (1 + 0.12 / 12)12 –1
= (1 + 0.01)12 –1
= 1.12683 –1
= 0.12683
= 12.68%.
Another example: A 12% nacq rate converts to an effective rate as follows:
ire = (1 + ir
n / t)t – 1
= (1 + 0.12 / 4)4 –1
= (1 + 0.03)4 –1
= 1.12550 –1
= 0.12550
= 12.55%.
It will be evident that a 12% naca rate will be equal to an effective rate of 12%.
Thus, the more interest periods involved, the higher the effective rate will be.
9. COUPON RATE
Ninetynine per cent of bonds and longterm NCDs have a coupon rate printed
on the face of the certificate (or on the computer generated letter / printout in the
age of dematerialisation). This is the fixed rate of interest payable to the
registered holders of the bonds on the specified interest payment dates. The
payment dates may be monthly, quarterly, semiannually or annually. Semi
annually is the most common.
The origin of the word coupon is the bond certificates of decades ago which had
coupons attached. These bonds were issued to bearer and they had a coupon
for each interest payment. On interest dates the holder detached the relevant
coupon and presented it to the issuer (mainly the government) for payment of the
interest.
The modern equivalent of the physical coupon is the coupon rate (cr) printed on
the face of the certificate. For example, a LCC 1 million bond may have a coupon
of 12.0% pa and interest payment dates of 30 June and 30 December. On these
interest dates an amount of LCC 60 000 would be paid to the registered holders:
15
Interest payable = (cr / not) x LCC 1 000 000
= 0.12 / 2 x LCC 1 000 000
= LCC 60 000.00.
The bonds that do not have a coupon rate are:
Variable rate bonds (such as the inflationlinked bonds).
Zero coupon bonds.
Islamic bonds.
Zero coupon bonds are issued for periods of longer than a year and only the
nominal / face value (FV) is payable on the maturity date. This of course means
that zero coupon bonds are issued at a discount, and that the interest earned =
FV – PV.
It is to be noted that the coupon earned by the holder is not necessarily the
actual rate that s/he is earning. The coupon rate is the rate earned only if the
bond is issued or trades at a price of 1.0 or 100%. In most cases bonds are
issued and trade at a price premium (for example 102.4%) or a price discount
(for example 92.8%). We take this further in the next section.
10. PRICE OF A SECURITY: THE PRINCIPLE
The price of a fixed interest rate security is inversely related to the market
interest rate for the security. The best example to demonstrate this is that of a
bond that has a fixed rate payable but has no maturity date: the perpetual bond.
The price of this bond is:
Price = cr / ir
where
cr = coupon payment (assumed to be annual)
ir = interest rate (at which the perpetual bond trades).
It should be clear that when cr = ir, the price is 1.0 or 100%. However, in the case
of a perpetual bond that has an annual coupon of 10% pa, but is trading at 9%
pa, the price is:
Price = 10% / 9%
= 1.1111111.
The principal at work here is that when the market rate for the perpetual bond
falls from 10% pa to 9% pa, the buyers are prepared to earn 9% pa in perpetuity.
This means that they are prepared to pay a price for the security that will yield
them 9% pa. On a LCC 1 million nominal / face value perpetual bond the annual
16
income is LCC 100 000 (= cr = 10% pa). Thus, the buyers will be prepared to pay
LCC 1 111 111.11 for the bond:
Consideration = (10% / 9%) x LCC 1 000 000
= 1.1111111 x LCC 1 000 000
= LCC 1 111 111.11.
Another example is called for:
Security = government bond
Nominal value = LCC 1 000 000
Coupon rate (cr, ie fixed rate for the period) = 15% pa
Coupon payable = in arrears, on maturity
Term to maturity = 365 days
Market rate = 15% pa.
The price of the bond on the issue date is 1.0 or 100%, ie the investor pays LCC
1 000 000 for the bond. If s/he holds the security for the period of 365 days, s/he
will earn the coupon:
Coupon = LCC 1 000 000 x 15.0 / 100
= LCC 1 000 000 x 0.15
= LCC 150 000.
However, if the interest rate for the bond in the secondary market falls to 7.5% pa
on the same day (the day of issue), the price of the bond will be 2.0 or 200%.
This is because there are buyers that are willing to accept a fixed interest rate of
7.5% pa for the period. (Remember that the coupon rate of 15% pa does not
change.) In terms of the formula shown above the price of the bond changes to:
Price = cr / ir
= 15.0% / 7.5%
= 2.0.
The consideration payable is:
Consideration = nominal value x price
= LCC 1 000 000 x 2.0
= LCC 2 000 000.00.
It will be evident that the buyer will earn:
Rate earned = coupon / LCC 2 000 000
= LCC 150 000 / LCC 2 000 000
= 0.075
= 7.5% pa.
17
This is the market rate at which s/he bought the bond. This demonstrates the
principle: price and market rates are inversely related. We now turn to reality:
most bonds worldwide have longer terms and have multiple regular interest
payments.
11. PRICE OF A SECURITY: MULTIPLE FUTURE CASH FLOWS
AND YIELD TO MATURITY
As said, the vast majority of bonds have longer terms to maturity (up to 30 years)
and have multiple and regular (usually twice pa) coupon interest payments. It is
best to elucidate with an example However, before we do so we need to
introduce the concept yield to maturity (ytm). Although in the bond markets of the
world the brokerdealers refer to a “rate” on a longterm security, they are
actually referring to its ytm.
Ytm is a measure of the rate of return on a bond that has a number of coupons
paid over a number of years and a face value payable at maturity. It may also be
described as the price that buyers are prepared to pay now (PV) for a stream of
regular payments and a lump sum at the end of the period for which the bond is
issued. It is an average rate earned per annum over the period.
Formally described, the ytm is the discount rate that equates the future coupon
payments and principal amount of a bond with the market price. Another way of
stating this is: the price is merely the discounted value of the income stream (ie
the coupon payments and redemption amount), discounted at the market yield
(ytm).
The following example will illuminate the PVFV of a longer term bond or NCD.
We choose a 3year maturity and annual interest payments to elucidate:
Settlement date: 30 / 9 / 2014
Maturity date: 30 / 9 / 2017
Coupon rate (cr): 9% pa
Nominal / face value: LCC 1 000 000
Interest date: 30 / 9
Ytm (ie market rate) 8% pa (payable annually in arrears).
The cash flows and their discounted values (the ytm is used) are as shown in
Table 1.
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The value now of the bond is LCC 1 025 770.96, and the price of the bond is
1.02577096. The price is calculates as follows:
Price (PV) = [cr / (1 + ytm)
1] + [cr / (1 + ytm)2] + [cr / (1 + ytm)3] +
[1 / (1 + ytm)
3]
where:
cr = coupon rate pa (expressed as a fraction of 1)
ytm = yield to maturity (expressed as a fraction of 1).
Using the same numbers as above (coupon rate = 9% pa, ytm = 8% pa):
Price (PV) = (0.09 / 1.08) + (0.09 / 1.166400) + (0.09 / 1.259712) +
(1 / 1.259712)
= 0.08333333 + 0.07716049 + 0.0714449 + 0.79383224
= 1.02577096.
It will be apparent that the coupon rate (0.09) for the periods and the face value
(1) that takes place at maturity (all FVs) are discounted at the ytm to PV.
Because the coupon rate is higher than the ytm, the price is higher than 1 (a
premium to par). Where the coupon rate is equal to the ytm (assume 9% pa) the
price is equal to 1 (par):
Price (PV) = (0.09 / 1.09) + (0.09 / 1.1881) + (0.09 / 1.295029) +
(1 / 1.295029)
= 0.082569 + 0.075751 + 0.069497 + 0.772183
= 1.000000.
As noted, when the coupon rate is lower than the ytm (assume coupon rate = 9%
pa, ytm = 11% pa), the price is lower than 1 (ie at a discount to par):
TABLE 1.1: CASH FLOWS AND DISCOUNTED VALUES
Date Coupon
payment (C)
Nominal / face
value
Compounding
periods (cp)
Present
value
C / (1 + ytm)cp
30/9/2015
30/9/2016
30/9/2017
30/9/2017
LCC 90 000
LCC 90 000
LCC 90 000




LCC 1 000 000
1
2
3
3
LCC 83 333.33
LCC 77 160.49
LCC 71 444.90
LCC 793 832.24
Total LCC 270 000 LCC 1 000 000 LCC 1 025 770.96
C = coupon payment. cr = coupon rate. cp = compounding periods (years).
19
Price (PV) = (0.09 / 1.11) + (0.09 / 1.232100) + (0.09 / 1.367631) +
(1 / 1.367631)
= 0.081081 + 0.073046 + 0.065807 + 0.731191
= 0.951125.
The inverse relationship between ytm and price is clear. This is because the ytm
is the denominator in the formula. Thus, if the ytm falls, the price of the bond
rises. It follows that if the ytm increases the price falls. Another way of seeing this
phenomenon is the logic of: As the ytm rises the future cash flows are worth less
when discounted to present value, pulling down the price.
In reality bonds are slightly more complicated but the principle remains the same.
The majority (by far) of bonds issued in the bond market have coupons that are
payable sixmonthly in arrears, and they are issued and traded for periods that
are broken, ie issues and secondary market settlement dates are between
interest payment dates.
In the case where interest payments are made sixmonthly in arrears (ignoring
settlement between interest payment dates), the coupon rate is halved and the
compounding periods are doubled (assume a threeyear bond):
Price (PV) = [(cr / 2) / (1 + ytm / 2)
1] + [(cr / 2) / (1 + ytm / 2)
2] +
[(cr / 2) / (1 + ytm / 2)
3] + [(cr / 2) / (1 + ytm / 2)
4] +
[(cr / 2) / (1 + ytm / 2)
5] + [(cr / 2) / (1 + ytm / 2)
6) +
[1 / (1 + ytm / 2)6].
The bond formula is usually written as:
n
Price = Σ [cr / (1 + ytm)t] + [1 / (1 + ytm)n]
t=1
where
cr = coupon rate (cr / 2 if sixmonthly)
ytm = yield to maturity (ytm / 2 if sixmonthly)
n = number of periods (years x 2 if sixmonthly).
12. OTHER ISSUES AND TERMINOLOGY RELATED TO
INTEREST RATES
12.1 Basis points, percentage points
It has been uttered that “interest rates have increased by one per cent”, or “the
central bank cut rates by a half per cent”. Both expressions are incorrect,
because a percentage change implies the change in interest rates from one level
20
to another level. For example, if a rate of interest changes from 10.5% pa to
11.5% pa, then the percentage increase is 9.52% [(11.5 / 10.5) – 1) x 100], not
1%.
The correct terminology is the interest rate increased by 100 basis points or 1
percentage point. The basis point concept was developed to explain small
movements in interest rates. Thus, a basis point is equal to 1 / 100 of a
percentage point.
12.2 Floating rate and fixed rate securities
We saw above that a fixed rate security (aka fixed interest security) is a security
(evidence of debt), which carries a fixed rate of interest, ie the rate of interest
payable by the issuer (borrower) remains unchanged throughout the life of the
security, irrespective of the rate at which the security trades in the secondary
market.
On the other hand, a floating rate security is a security on which the rate of
interest changes daily or less frequently (depending on the deal). An example of
a true floating rate security is a “call deposit”, ie the interest rate on the deposit
can change daily. Another example is a 3year corporate bond issued at the 91
day TB rate + 100 basis points. It reprices every 91 days with the issue of new
91day TBs. In other words a floating rate debt security is a debt on which the
rate is benchmarked on a wellpublicised rate.
Apart from the TB rate, the most often used benchmark rates are:
A wellpublicised interbank rate. An example is the UK LIBOR rate
(London interbank offer rate), which in theory can change daily.
The prime lending rate (PR) of the banks (aka base rate, bank rate, etc).
The policy interest rate (PIR) of the central bank (aka discount rate, repo
rate, bank rate, key interest rate, etc).
12.3 Primary and secondary market rates
As said earlier the primary market is the market for the issue of new securities
and the secondary market is the market for the trading of existing securities (ie
securities that are already in issue). The rate in the primary market can be called
an issue rate or a primary issue rate, but usually the former. The rate in the
secondary market is called the secondary market rate or just the market rate
(usually the latter). For example, when a new government bond is issued it is
issued at an issue rate. When it trades in the secondary market it trades at the
market rate (the ytm).
Primary market rates are unimportant (in terms of analysis), compared with
secondary market rates, because they are issue rates that applied at a particular
21
time, and they in any case are established with reference to the secondary
market rates. Also, issues of particular securities are not made on all days.
Secondary market rates are important and studied by analysts and academics
because they are discovered / established in some cases every second of the
day. Because of this they provide time series’ of various rates.
However, this does not apply to certain high profile primary rates that originate in
markets that do not have secondary markets: Examples are the call money rates
paid by banks, the prime lending rate (PR) of banks, the mortgage rate of banks
and the policy interest rate (PIR) of the central bank (which of course are all
closely linked).
12.4 Nominal value and maturity value
We have covered nominal value and hinted at the concept maturity value. We
need here to distinguish between money market (all shortterm) and bond market
(marketable longterm) securities. As we have seen, the plain vanilla bond is one
that pays a coupon periodically for a number of years on an amount. This amount
is the nominal value (aka face value) of the bond. An example will elucidate:
Security = government bond
Nominal value = LCC 1 000 000
Coupon rate (ie fixed rate for the period) = 15% pa
Coupon payable = annually in arrears
Term to maturity = 5 years
Market rate = 10% pa.
When this bond has less than 12 months to maturity, on maturity it will pay LCC
1 000 000 + the coupon amount of LCC 150 000 = LCC 1 150 000 to the holder.
This is the maturity value (MV). If someone buys this bond when it has 85 days to
maturity at a rate (rate, no longer ytm, applies here) of 9.5% pa, she will pay:
Consideration (PV) = MV (= FV) discounted at 9.5% pa
= LCC 1 150 000 / [(1 + (0.095 x 85 / 365)].
= LCC 1 150 000 / 1.02212329
= LCC 1 125 108.89.
Because this bond has less than a year to maturity, it falls into the money market,
and is also called an interest addon security. In the money market here are two
main types of fixed interest securities (they have one interest payment):
Interest addon securities.
Discount securities (covered later).
22
An example of the former is the bond covered above. Another is the NCD. A
buyer of a new NCD (ie a depositor) will deposit at the bank LCC 1 000 000 at a
rate of 8.25% pa for 182 days. The maturity value (MV = FV) of the NCD is:
MV (FV) = PV x [1 + (0.0825 x 182 / 365)]
= LCC 1 000 000 x 1.04113699
= LCC 1 041 136.99.
If this NCD is sold in the secondary market after 10 days (ie has 172 days to
maturity) at 7.5% pa, the calculation of the consideration (PV) is done according
to the PVFV formula presented above in the case of the short bond:
Consideration (PV) = MV (= FV) discounted at 7.5% pa
= LCC 1 041 136.99 / [(1 + (0.075 x 172 / 365)].
= LCC 1 041 136.99 / 1.03534247
= LCC 1 005 596.72.
12.5 Yield rate and discount rate
So far we have worked with yields, which are the actual rates of return on
securities. A variation is ytm in the case of bonds, which is an average yield /
return.
We said above that in the money market we find discount securities. An example
is the TB. A TB with a nominal / face value of LCC 1 000 000 matures at LCC
1 000 000 (= FV), but it is issued and traded at a discount rate. This TB can for
example trade at LCC 950 000 (depending on the discount rate), calculated
according to (dr = discount rate; d = days to maturity):
PV = FV x [1 – (dr x d / 365)].
If a LCC 1 000 000 (= FV) TB has 91 days to run and trades at 11.0% pa, its
consideration is:
Consideration (PV) = LCC 1 000 000 x [1 – (0.11 x 91 / 365)]
= LCC 1 000 000 x (1 – 0.02742466)
= LCC 1 000 000 x 0.97257534
= LCC 972 575.34.
From the above it will be apparent that there is a fundamental difference between
a discount rate and a yield rate and therefore between a discount amount and a
yield amount. A yield amount is based on the PV, and the FV is the sum of the
two. The discount amount, on the other hand, is based on the FV, and the PV is
the difference between the two. It follows that the yield rate of interest is always
expressed as a percentage of the PV, while the discount rate is expressed as a
percentage of the FV.
23
It will be evident that the FV of the LCC 1 000 000 TB is also a MV. Thus, we can
arrive at the same consideration as above if we convert the discount rate to a
yield rate using the conversion formula:
ir = dr / [1 – (dr x d / 365)]
= 0.11 / [1 – (0.11 x 91 / 365)]
= 0.11 / (1  0.02742466)
= 0.11 / 0.97257534
= 0.1131
= 11.31%.
The proof (refer to the last consideration calculation above):
Consideration = MV / [1 + (0.1131 x 91 / 365)]
= LCC 1 000 000 / 1.02819753
= LCC 972 575.34.5
In conclusion, the yield interest rate to discount rate conversion formula:
dr = ir / [1 + (ir x d / 365)].
12.6 Riskfree rate
Some textbooks are confusing on the subject of the riskfree rate (rfr). Some
define it as the rate on 3month TBs, while others say it does not exist. Our view
is that there are many rfrs and they can be found on the government security
(bonds and TBs) yield curve. The government security yield curve is a snapshot
of all rates on government securities in issue (ytms on bonds and yields on TBs)
at a specific point in time. A yield curve, also called the term structure of interest
rates, presents the relationship between term to maturity and rates at a point in
time. It is discussed in detail later.
Why is the rfr important? It is a rate that is used in many financial market
calculations, especially in the derivative markets. It also represents the basis of
the rate that an investor should accept on risky assets (ie nongovernment
securities, such as shares, corporate bonds, etc). Thus, the rate on a risky
security, aka required rate of return (rrr), is equal to the rfr plus a risk premium
(rp):
rrr = rfr + rp
The investor has to decide what the rp should be. There are many studies on the
size of the rp, such as the capital asset pricing model (CAPM).
5 The cents’ numbers are the same when more decimals are used.
24
What does riskfree mean? Government securities are considered riskfree
because they have the ability to raise revenue (tax and issue securities), and
thus always service debt and honour maturities. As is well known, some
countries’ government securities are not riskfree, but such countries are few.
12.7 Bid and offer rates / prices and spread
There are two debt / deposit market types:
Order driven markets.
Quote driven markets.
In orderdriven financial markets, such as share exchanges, sellers or buyers
place orders to sell or buy shares with their brokers. In the age of share
exchanges’ Automated Trading Systems (ATS), the ATS’s central order book
arranges the sell and buy prices according to best price followed by the inferior
prices. Deals are struck by the ATS when the buy and sell prices coincide.
In the debt and deposit markets, the main market type is quotedriven (there is
an element of order placing), in the sense that the market (certainly in most bond
markets) is “made”. This means that market makers (usually the banks) quote
both buy and sell rates / prices simultaneously. They are the bid (the market
maker’s buying rate; the selling rate from the perspective of the client) and offer
rates (the market maker’s selling rate; the buying rate from the viewpoint of the
client). In some countries these rates are known as bid and ask rates.
The bid price is always lower than the offer price and the difference is called the
spread. The spread is the compensation for the market maker for the risk taken
in quoting firm bid and offer prices simultaneously. Firm means that the market
maker is prepared to deal at the prices quoted in a given volume (which
disclosed by the client). We will return to this issue later, as it is an important part
of price discovery of rates.
12.8 Nominal and real interest rates
Any interest rate published in the media is a nominal interest rate (nir). An
interest rate adjusted for inflation () is a real interest rate (rir), and it reflects the
true cost of borrowing / the true earning rate. The first person to “split” the rate is
Prof Irving Fisher, and the equation of the real interest rate, named for him
(Fisher equation), is:
rir = nir  .
25
The inflation rate used can be the current rate if it is low and has been level for
some time, or the expected rate if this is not the case. We will return to this issue
later.
13. REFERENCES
Faure, AP (2012 – 2013). Various which can be accessed at
http://ssrn.com/author=1786379.
Faure, AP (2013). Various which can be accessed at
http://bookboon.com/en/bankingfinancialmarketsebooks
Mishkin, FS and Eakins, SG, 2009. Financial markets and institutions. 6e.
Reading, Massachusetts: AddisonWesley.
Rose, PS and Marquis, 2008. Money and capital markets. 10e. New York:
McGrawHill Higher Education.
Saunders A and Cornett, 2012. Financial markets and institutions. 5e. New
York: McGrawHill Higher Education.
Santomero, AM and Babbel, DF, 2001. Financial markets, instruments and
institutions. 2e. Boston: McGrawHill/Irwin.